Willpower and the Optimal Control of Visceral Urges�

Emre Ozdenoreny Stephen Salanty Dan Silvermany

This Version: January, 2008

Abstract

Common intuition and experimental psychology suggest that the ability to self-regulate

(�willpower�) is a depletable resource. We investigate the behavior of an agent with limited

willpower who optimally consumes an endowment of a tempting consumption good or �cake�

over time. We assume that restraining consumption of cake below what maximizes immediate

grati�cation requires willpower. Any willpower left over may be valuable in controlling other

urges. Willpower thus links otherwise unrelated behaviors requiring self-control. Our model

predicts domain-speci�c time preference and a linkage between expenditure on nontempting

goods and the value of self control in regulating other activities. It also shows that short-

sighted behavior by the poor may be attributable to their lack of wealth rather than their tastes

or skills. Unlike someone with perfect self-control, an agent with limited willpower will almost

never perfectly smooth his consumption, even when it is feasible to do so. Our model provides an

explanation for negative time preference, a phenomenon usually attributed to direct utility from

beliefs (anticipation). At the same time, it provides a novel explanation for self-commitment,

intertemporal preference reversals, and procrastination.

�We thank Roy Baumeister, Gérard Gaudet and Miles Kimball for their important contributions to this paper.

We also thank Roland Benabou, Doug Bernheim, Drew Fudenberg, Kai-Uwe Kuhn, Yoram Halevy, David Laibson,

David Levine, Robert Mendelsohn, Muriel Niederle, Andrew Postlewaite, Matthew Rabin, Lones Smith, Joel Sobel,

Kathleen Vohs, Itzhak Zilcha, and participants in seminars at Harvard, IAS/Princeton, SITE, Calgary, Central

Florida, Michigan, Milano-Bicocca, Montreal, Penn and Pittsburgh for their helpful comments on a previous draft.

Silverman gratefully acknowledges the support of the Institute for Advanced Study.yDepartment of Economics, University of Michigan, 611 Tappan St., Ann Arbor, MI 48109-1220.

1 Introduction

Common patterns of intertemporal choice that are inconsistent with standard models have

inspired a literature on the economics of self-control. Tendencies to seek self-commitment, to

procrastinate, and to seize small immediate rewards despite their important future costs, have

motivated studies of quasi-hyperbolic time discounting (e.g., Laibson 1997, O�Donoghue and Rabin,

1999), temptation costs (e.g., Gul and Pesendorfer 2001, 2004), and con�icts between selves or

systems (e.g., Thaler and Shefrin, 1988; Bernheim and Rangel, 2004; Fudenberg and Levine, 2006).

These models are consistent with a great deal of experimental evidence, and have been fruitfully

applied to a number of economic problems ranging from portfolio choice to labor supply to health

investment.

Our paper is focused on an aspect of self control that has received less attention in the literature:

the variation in an individual�s exercise of self restraint across time and circumstances. Speci�cally,

we are concerned with anecdotal and laboratory evidence that the exercise of self control at one

time, or in one domain, leaves a person less able or willing to exert self-restraint at another time,

or in another domain. This evidence suggests that the exercise of self control draws on a limited

and fungible cognitive resource �a resource that is often called willpower.

Anecdotes consistent with limited willpower are common. Many who resist unhealthy food

and fruitless websur�ng all day, and who might prefer to go to bed early after a light dinner, �nd

themselves up late, watching T.V. and gorging on junk food. Dieters often maintain their discipline

for short periods but �nd such self-restraint unsustainable over the long term. Pro�igate spending

or drinking to excess is frequently the �reward�for a hard week at work or home.

Experimental psychology (Baumeister et al., 1994; Baumeister and Vohs, 2003) has gone beyond

such anecdotes and has demonstrated that individuals depleted by prior acts of self-restraint tend

to behave later as if they have less capacity for self-control. The typical experiment has two phases.

Every subject participates in the second phase but only a randomly chosen subset participates in

the �rst, with the remainder serving as a control group. In the �rst phase, subjects are asked to

perform a task that is meant to deplete their willpower; in the second phase, their endurance in an

unrelated activity requiring self control is measured.1 Subjects who participate in the �rst phase

display substantially less endurance in the second phase. This apparent link between the exercise

of self-control in one activity and later self-discipline in another has been observed repeatedly, with

many di¤erent manipulations and measures of self-regulation (Baumeister and Vohs, 2003, Vohs

1For example, in the �rst phase subjects have been asked to resist eating tempting foods, to resist drinking liquids

when thirsty, and to inhibit automated/habitual behaviors such as reading the subtitles of �lm or �reading�the color

of the ink that a word is written in, rather than the word itself.

1

and Faber, 2004). Two experiments (Vohs and Faber, 2004 and Dewitte et al., 2005) show that

willpower depletion and prior cognitive loads a¤ect subsequent economic behavior.2 Individual ex-

periments have weaknesses, but we view the collection of these experimental �ndings as reinforcing

the intuitive notion of willpower as a depletable cognitive resource.

In this paper we investigate how willpower limitations a¤ect economic behavior. To do so, we

add a willpower constraint to the simplest model of intertemporal choice: the canonical cake-eating

problem. In our formulation, moderating consumption of tempting goods requires willpower; the

greater restraint the consumer exercises, the faster his willpower erodes.3 We also take account of

other activities besides intertemporal saving that require willpower: cramming for exams, training

for performances or competitions, maintaining a diet, or preparing an important presentation.

While our model of limited willpower is simple, it generates a rich set of implications. We �nd

that a willpower-constrained consumer would behave in ways that seem inconsistent with having

a single rate of time discount. He might, for example, exhibit considerable patience when it comes

to work e¤ort and, at the same time, appear myopic when it comes to the long-term consequences

of food and drink. His time preference would thus appear domain-speci�c.

In addition, a willpower-constrained consumer who is poor might appear more impatient than

his richer counterpart even though they are identical in every respect besides their initial wealth.

The poorer consumer would deplete his paycheck sooner and might also devote less willpower to

resisting other temptations. Thus, behavioral di¤erences between rich and poor that are often

attributed to di¤erences in rates of time discount or self-control skills may also be due simply to

di¤erences in wealth.

When a consumer is willpower-constrained, even his demand for conventional (i.e. nontempting)

goods will be a¤ected. If, for example, the value of willpower allocated to regulating alternative

activities increases, the consumer will use less willpower to limit his intake of tempting goods and,

as he spends more on temptations, spend less on conventional goods.

2 In Vohs and Faber (2004), subjects who were willpower-depleted purchased a wider assortment of merchandise

and spent a larger portion of their experimental earnings than the control group. The experiments of Dewitte et

al. should be distinguished from research that showed the e¤ects of cognitive loads on contemporaneous choices. In

these latter studies, respondents were more likely to choose cake over fruit (Shiv and Fedorikin, 1999) or a smaller

earlier reward over a larger later one (Hinson, et al. 2003) when simultaneously asked to perform a memory task. In

contrast, Dewitte et al. (2005) found that even when such memory tasks were performed prior to the consumption

decision, they a¤ected the choice of how much candy to eat.3Thus, we mainly treat willpower as a resource that is depleted by resisting tempting consumption. Experiments

indicate that willpower may also be drained by other activities such as inhibiting automatic responses or bearing

cognitive loads. Our model accommodates these causes of depletion either as anticipated uses of willpower that come

after consumption choices, or as unanticipated shocks to the willpower stock.

2

Finally, limited willpower has important consequences for preferences over the timing of con-

sumption. The hallmark of the canonical model, intertemporal smoothing, virtually disappears

when a willpower constraint is introduced. Even when the agent has enough willpower to smooth

consumption perfectly over the entire time horizon, he is almost always better o¤ foregoing this

option and using more willpower to regulate alternative activities.4 Indeed, as we show, under a

willpower constraint it may even be optimal to consume at an increasing rate. Others have at-

tributed such negative time preference to direct utility from beliefs (anticipation). Our analysis

suggests it may also re�ect self-control problems.

The model presented here di¤ers from the bulk of the self-control literature by accommodating

an e¤ect of previous acts of self-restraint on current behavior that requires self-control. Models of

hyperbolic discounting (Laibson 1997, O�Donoghue and Rabin, 1999) or temptation costs (Gul and

Pesendorfer 2001), for example, would predict that, so long as the agent faces the same choice set,

the history of his previous choices would not matter. Models of addiction, habituation, or cues,

(Laibson, 2001, Gruber and K½oszegi, 2001, Gul and Pesendorfer, 2007, Bernheim and Rangel, 2004)

allow the history of choices to a¤ect behavior, but in these models the consumption of a good today

a¤ects later tastes for the same good whereas in our model, the willpower constraint links (induced)

preferences across all forms of tempting consumption. The model in Benabou and Tirole (2004)

also links past behavior to future choices as agents with hyperbolic discounting attempt to develop

self-reputations for being patient.5 However, in Benabou and Tirole (2004)� unlike our model� the

successful exercise of self-control in one domain increases the likelihood of later self-control. We

defer further discussion of the relation of our model to other particularly relevant contributions of

the self-control literature (Fudenberg and Levine (2006), and Loewenstein and O�Donoghue (2005))

until the implications of our model have been presented.

The rest of the paper proceeds as follows. In section 2 we present the primary elements of our

model. In the �rst part of that section, we focus on a particularly simple case of our model in order

to highlight the novel linkages in economic behavior induced by a willpower constraint. Because the

willpower constraint does not induce time preference in this case, we can show many consequences

of limited willpower using elementary calculus. In the remainder of the section, we consider the

4Allowing willpower to have alternative uses has other implications. For example, using the typical two-stage

protocol, Muraven (1998) �nds that a given �rst-stage depletion activity has less of an e¤ect on second stage self-

control when subjects are paid more for exerting that latter control. This �nding is consistent with a model where

willpower has alternative uses. If we �x the marginal value of willpower remaining after the experiment, and increase

the incentive for exerting self-control in the second phase activity, then the agent will optimally reallocate willpower

to that second phase self-control activity and leave less in reserve for self-regulation after the experiment.5Agents build self-reputations because the model allows uncertainty about preferences and imperfect recall of both

preferences and choices.

3

more general problem and discuss the rich patterns of time preference that the willpower constraint

can induce. In section 3 we discuss further how our theory relates to other models of self control.

In section 4 we consider two extensions of our analysis: to the case where an individual is uncertain

about the size of his willpower stock and to the case where the exercise of self control depletes

willpower over the short term but builds it over the longer term. In section 5 we conclude.

2 A Model of Limited Willpower

To investigate the consequences of limited willpower in the simplest intertemporal model, we

consider the canonical cake-eating problem where a consumer chooses his consumption path c (t)

to maximize his discounted utility U (c (t)) over a �nite horizon. We denote the size of the cake at

time t as R(t) and assume that R(0) is given. There is no return to saving in the model; thus at

time t the cake declines at rate c (t) :

We depart from the canonical model by assuming that the cake is tempting and that the

agent, initially endowed with willpower stock W (0); depletes his willpower when he restrains his

consumption. We denote the rate of willpower depletion at time t as f(W (t); c(t)): Thus depletion

may depend both on the rate of consumption (self restraint) and on remaining willpower reserves.

We assume that instantaneous utility is maximized at a consumption rate �c and no willpower

is used to consume at that rate. To restrain consumption below that rate, however, does require

willpower. We assume that the more the agent restrains his consumption, the faster he depletes his

willpower reserves. We also allow that the same act of self-restraint may result in (weakly) faster

depletion of willpower if the agent�s reserves of willpower are lower. Formally, we assume that f is

strictly decreasing (fc < 0) and weakly convex (fcc � 0) in c and weakly decreasing inW (fW � 0).6

We assume U (0) = 0 and, for c 2 (0; �c); that utility is strictly concave in c, strictly increasingand twice di¤erentiable. Our focus is on non-addictive goods like those captured in experiments

by Baumeister and colleagues, so we assume that U (�) is a function only of contemporaneousconsumption.

An important feature of willpower depletion is that as long as even a morsel of tempting cake

remains, restraining consumption requires willpower; but when there is nothing left to eat, resist-

ing eating it requires no willpower. We refer to this feature of the willpower technology as the

�fundamental discontinuity of willpower depletion�and take account of it in our formulation.7

6For completeness, we assume that the cross e¤ects are positive, fcW � 07This discontinuity would persist even if f (�) were a function of R (t) so long as lim

R(t)!0f (�) > 0: It would simplify

our analysis if the discontinuity disappeared, but there is no reason to expect that it does.

4

We assume that any willpower remaining after moderating intertemporal consumption is ap-

plied to restrain urges in alternative activities. This willpower �bequest�does not generate utility

directly. But it does alter the feasible set of choices in the alternative activities and therefore does

indirectly determine the utility generated from the alternative activities. We denote this indirect

utility function as m(�):The agent cannot choose a consumption path which results in negative willpower. From the set

of consumption paths which maintain nonnegative stocks of both willpower and cake, the consumer

chooses the one which maximizes the sum of his discounted utility of consumption and the value

of the willpower bequeathed to the regulation of his alternative activities.8

If a consumer with initial willpowerW and initial cake R chooses his consumption path c(t) � 0optimally, he maximizes:

V (0) =

Z T

0e��tU [c (t)] dt+ e��sm (W (s))

subject to�R (t) = �c (t) (P1)�W (t) =

��f (W (t) ; c (t)) if R (t) > 0

0, otherwise

R (T ) � 0; W (T ) � 0

R (0) = R � 0

W (0) =W � 0;

where � is the subjective rate of time discount and s = sup ft 2 [0; T ] : R (t) > 0g.9 The functiondescribing changes in the stock of willpower jumps when the cake is exhausted, re�ecting the

fundamental discontinuity of willpower depletion discussed above.

As discussed below, the inclusion of a willpower constraint a¤ects consumer behavior in two

ways: it induces linkages between behaviors in di¤erent domains and it can induce time preference

in consumption. To study these e¤ects separately, we investigate �rst a particularly simple case

where willpower induces linkages but not time preference and then move on to the more general

8As the various metrics used in experiments re�ect, willpower may be measured in many ways. A natural concern is

that the model�s predictions about optimal consumption depend on how willpower is measured. However, the optimal

consumption path does not depend on how willpower is measured. The reason is simple. Every consumption path

that is feasible when willpower is measured in one way is also feasible when it is measured in another. Since utility

depends only on the consumption path chosen, the same consumption path is optimal regardless of how willpower is

measured. We thank Doug Bernheim for raising this concern about measurement.9We have assumed, for convenience, that the value of the willpower bequest to alternative activities is realized

either when the cake is exhausted (s) or at the end of the horizon (T ) ; whichever comes �rst. This assumption is not

material to our results.

5

case where willpower concerns can also induce time preference.10

2.1 Induced Linkages

We begin our analysis by studying optimal choice in the particularly simple case where the

rate of willpower depletion depends only on the level of consumption. Formally, we assume that

f(W (t); c(t)) = g(c(t)).11 Moreover, we assume that additional willpower devoted to alternative

activities has a constant marginal value (m0(W (t)) � m). To isolate the e¤ects of limited willpower,we assume there is no time discounting (� = 0).

Under these assumptions, it is optimal to consume the tempting good at a constant rate as long

as cake remains. To see why this is true, suppose the contrary. That is, suppose consumption were

larger at one time than at another when cake remains. Because utility is strictly concave in tempting

consumption, marginally reducing the larger consumption while marginally increasing the smaller

one by an o¤setting amount would strictly increase utility from consumption without violating the

cake constraint. Whether this perturbation strictly increases overall utility depends on its e¤ect on

willpower. Recall, however, that willpower depletion is independent of the willpower stock and is a

decreasing, convex function of consumption. It follows that the proposed perturbation will actually

make more willpower available for use elsewhere: increasing the lower level of consumption releases

more willpower than is depleted by reducing the larger consumption by an o¤setting amount. At

all other times, when consumption is left unchanged, willpower depletion is unchanged (because

fW = 0). Thus the perturbation releases additional willpower to be used in the alternative activities

and thereby generates another gain in utility. So if the original program was feasible, the perturbed

program will also be feasible but it will yield higher utility. Hence, consumption must be constant

as long as cake remains.12

The preceding argument shows that consumption is constant when it is positive. In principle,

however, cake may be exhausted prior to T , after which consumption jumps to zero. A perturbation

that further smooths consumption in this situation has a decidedly di¤erent e¤ect than the one

described above. In this situation, marginally reducing consumption in the phase when it is positive

10Because the willpower depletion technology is discontinuous, problem (P1) is somewhat nonstandard. Therefore,

in each investigation, we will solve problem (P1) by analyzing a simpler, conventional problem with the same solution.

Details regarding the equivalent problem are provided below in Section 2.2.11Two prominent special cases of this formulation are g (c (t)) = k �(�c�c(t)) and g(c(t)) = k �(U(�c)�U(c(t)); where

�c is the lowest rate of consumption that maximizes instantaneous utility. In the former case, the rate of willpower

depletion is proportional to the di¤erence between the agent�s ideal and his actual rate of consumption; in the latter

case, it is proportional to the di¤erence between the utilities generated by these two rates of consumption.12See footnote 19 for a more formal proof of these arguments.

6

and marginally increasing it in the phase when it is zero depletes willpower in both phases. This

occurs because of the fundamental discontinuity of willpower depletion: when no cake is available

no willpower is depleted, but once cake becomes available in the second phase, consuming a small

amount requires additional willpower. As the analysis below clari�es, it is not always optimal to

stretch the phase of constant consumption to the end of the horizon. Intuitively, when the return

to alternative uses of willpower is high, it will be better to complete consumption before time T and

save more willpower for other purposes. We will call the situation when the agent chooses to stretch

consumption to time T the �perfect-smoothing regime,�and call the alternative the �no-smoothing

regime.�

2.1.1 Simple Comparative-Statics

In the rest of this section, we assume that the agent has enough cake or willpower to implement

perfect smoothing if he so chooses.13 Then the payo¤ from consuming at constant rate c and

exhausting the cake at time s � T is sU(c)+m[W�sg(c)]; where s = Rc and s 2 [0; T ]: Provisionally

substituting Rs for c we obtain a continuous, strictly concave objective function of one variable, s:

The agent�s problem is then equivalent to choosing the time to exhaust the cake, s; to solve:

maxssU(

R

s) +m[W � sg(R

s)]

It is straightforward to verify that an optimum exists and that, since the maximand is strictly

concave, any solution to the �rst-order condition is optimal. The optimum never occurs at s = 0

but may occur at s < T or at s = T . We consider each case in turn.

No-smoothing Regime When no smoothing is optimal, the �rst order condition implies:

U(R

s)� U 0(R

s)R

s�mg(R

s) +

R

smg0(

R

s) = 0: (1)

The objective function is strictly concave, so the left-hand side of (1) is strictly decreasing in s: If

it did not decline to zero for s < T; then perfect smoothing (s = T ) would be optimal.

To determine the rate of consumption in the no-smoothing regime, we rewrite the �rst-order

condition in terms of the constant consumption rate c:

U(c)� cU 0(c)�mg(c) + cmg0(c) = 0: (2)

Notice that the rate of consumption in the no-smoothing regime is determined entirely by the

value of additional willpower in the alternative activities (m). Hence, if an agent in this regime

13That is, W � Tg(RT) > 0. The case where he lacks enough cake or willpower to implement perfect smoothing is

equally tractable but less interesting.

7

began with larger initial willpower, he would optimally consume at the same rate and, with a cake

of unchanged size, would exhaust it at an unchanged time (s), bequeathing all of the additional

willpower to the alternative activities.

Turning next to changes in initial cake sizes, suppose someone in the no-smoothing regime began

with a marginally larger initial cake. Then his rate of consumption would not change (since m has

not changed); but since he would take longer to exhaust his larger cake later (s increases) he would

have less willpower left for the alternative activities.

Now consider changes in the value (m) of willpower applied to alternative activities. Suppose

someone in the no-smoothing regime began with a marginally higher m. It is straightforward to

verify that the rate of consumption is a strictly increasing function of m.14 Hence, the optimal rate

of consumption would increase in response and, given that the cake size is unchanged, it would be

exhausted sooner.

Perfect-smoothing Regime In contrast, when perfect-smoothing is optimal, no behavior changes

in response to an exogenous increase in the value of willpower in the alternative activities (m). For,

perfect smoothing requires that the cake still be exhausted at T and, given a cake of unchanged

size, the consumer would devour it at the same rate and would therefore have the same amount of

willpower to bequeath. If someone who would perfectly smooth received marginally more willpower,

he would not alter his consumption path or date of exhaustion but would simply use the additional

willpower in the alternative activities. If such a person instead began with a larger cake, then he

would consume it at a faster rate, exhausting it at an unchanged date (s = T ), and he would have

more willpower left over for the alternative activities. We summarize the comparative-static results

for each regime in Table 1 and then turn to the behavioral implications of these results.

2.1.2 Smoothing Consumption May Not Be Optimal

Consumption smoothing is the hallmark of the standard model with stationary utility and no

discounting. However, it does not survive the addition of willpower concerns when willpower is

needed not only to resist tempting consumption but also to regulate other urges. These urges may

include the temptation to slack o¤ at work or school, the urge to participate in risky �nancial

or sexual behaviors, or the urge to express anger or jealousy to co-workers, friends or family.

Suppose the return to the alternative activities is high enough, so that the agent is in the no-

14Totally di¤erentiating the �rst-order condition determining c in the no-smoothing regime, we conclude that

dc=dm = (g(c)� cg0(c)) = (c[mg00(c)� U 00(c)]) > 0, where the sign follows since g(�) is positive, strictly decreasing,and strictly convex while U is strictly concave.

8

c s W (s)

m + � +

�R 0 + �W 0 0 +

c s W (s)

m 0 0 0

�R + 0 +

W 0 0 +

No Smoothing Perfect Smoothing

Table 1: Comparative-Static Results in Each Region. Sign of changes in optimal consumption

(c), time to exhaust the cake (s), and willpower bequest W (s), resulting from changes in the return

to willpower applied to alternative activites (m), changes in initial cake size (R), or changes in

willpower (W ).

smoothing regime.15 If this agent had more willpower, he would use no more to restrain tempting

consumption; instead he would use the additional willpower to restrain urges in other activities.

(See the bottom rows in Table 1.) Therefore, if m is high enough, the agent would continue to

avoid perfect smoothing of consumption no matter how large is his initial stock of willpower.

2.1.3 Domain-Speci�c Time Preference

This simple model also shows how, for an agent with limited willpower, time preference may

di¤er sharply by decision domain. The individual may appear willing or able to postpone grati�-

cation in one set of activities and, at the same time, profoundly myopic about choices in another,

even though in fact he discounts time at a single rate (assumed to be zero here). Consider the

following concrete example. Suppose willpower is used both to regulate consumption and to exert

concentrated e¤ort on dull but professionally important tasks at work �tasks which provide only

longer-term rewards. If the return (m) to more concentrated e¤ort at work is high enough so that

the agent is in the no smoothing regime, an increase in m will result in the agent consuming his

tempting cake faster, exhausting it more quickly, and therefore having more willpower for work

related activities. (See Figure 1 and the top rows in Table 1.) Hence, the agent would appear at

the same time more myopic when it comes to consumption choices, but more forward-looking when

it comes to work.

Short-sighted consumption behavior and diligence at work might also occur if the individual

were truly myopic but simply enjoyed work. However, this explanation can be distinguished from

the e¤ects of willpower. In our model as the return (m) to additional willpower allocated to work

15To see which regime is optimal for given parameters m and R, set s = T in the left hand side of (1). If it is

nonnegative, then perfect smoothing is optimal; otherwise no-smoothing is optimal. Notice that, for any cake size,

no smoothing is optimal for any m above a critical level.

9

increases, the individual would appear increasingly impatient in his consumption and increasingly

disciplined in his regulation of alternative activities. In contrast, an increase in returns to e¤ort

at work would not a¤ect short-run time preference for consumption in a model that attributed

such domain-speci�c time preference to tastes for work or to domain-speci�c rates of time discount.

Thus, the limited-willpower model makes distinctive predictions about changes in self-restraint

exercised in one activity as a function of the return to exerting self-restraint in another.

2.1.4 Poverty A¤ects Patience

Willpower concerns also induce a distinctive linkage between poverty and patience. In the

standard view, poverty is partially caused by impatience: impatient agents are less willing to

postpone grati�cation and invest in human and �nancial capital, and thus they accumulate less

wealth.16

The willpower model suggests another relationship between poverty and patience. Consider two

agents with the same willpower reserves and self-control technologies, but one su¢ ciently poor (has

a small enough cake) that no smoothing is optimal and the other su¢ ciently rich such that perfect

smoothing is optimal. The poorer one may both �nish his �cake�sooner and have less willpower

left over for alternative activities than his richer counterpart. (See Figure 2.) The poor man thus

appears to lack self-control when compared with his wealthier counterpart; but the di¤erence in his

behavior is attributable entirely to his smaller paycheck. While impatience may lead to poverty as

present-oriented consumers forego opportunities to invest in wealth or human capital, the willpower

model suggests that the poverty of the poor may also make them impatient.

2.1.5 Willpower Concerns Even A¤ect Consumption of Nontempting Goods

So far, we have assumed that all consumption is tempting. In this subsection, we consider the

case where some goods, such as deodorant, oil changes, or plain yogurt, pose no meaningful temp-

tation. Although limiting consumption of such �nontempting goods�might require no willpower,

demand for them is nonetheless in�uenced by willpower concerns. For example, suppose a worker is

limiting his junk food and entertainment consumption over a �xed horizon while applying moderate

e¤ort to a project at work. He then learns that an in�uential manager has shown an interest in

the project and will closely monitor various aspects of it �an unusual opportunity to impress or

disappoint his superior. In response to this news, he will allocate more of his willpower budget

to work and less to restricting his consumption of junk food and his drinking at happy hours. To

�nance the additional consumption of these tempting goods, however, he would spend less of his

16See, e.g., Lawrance (1991).

10

budget on nontempting consumption. Thus, news of management�s interest in the project would

cause spending on nontempting goods to fall.

To see this formally, suppose that a �xed sum of money (I) can be spent either on goods requiring

willpower (intertemporal consumption) or on the consumption of nontempting goods. Suppose R

dollars is spent on intertemporal consumption of the tempting good (adjust units so that one unit

of cake costs one dollar) and RCG is spent on the set of conventional goods: R+RCG � I: Assumethat spending RCG on the conventional goods generates utility V (RCG), where the utility function

is strictly increasing and strictly concave.

Suppose that the agent is smoothing perfectly. We denote the agent�s payo¤ from activities re-

quiring willpower by J (R;m) � TU (R=T )+m�W � Tg (R=T )

�: Since U is strictly concave and g is

strictly convex, J is strictly concave in R: The agent will choose R to maximize V (I �R)+J (R;m) :At the optimum a dollar allocated to consumption of the tempting good and the nontempting good

must have the same value at the margin, that is, �V 0 (I �R)+JR (R;m) = 0: Since both V and Jare strictly concave in R, the second-order condition also holds; that is, V 00 (I �R)+JRR (R;m) < 0:To see how spending on the tempting good varies with the return to willpower used in the alter-

native activities, we take the total derivative of the �rst-order condition with respect to m to

obtain:dR

dm=

�JRm (R;m)V 00 (I �R) + JRR (R;m)

:

Since JRm (R;m) = �g0 (R=T ) > 0; we conclude dR=dm > 0:17 Thus when the return to exercising

self-control in alternative activities increases, the consumer devotes more of his budget to tempting

consumption leaving less for nontempting goods.

2.2 Induced Time Preference

Besides inducing linkages between otherwise unrelated behaviors, the introduction of a willpower

constraint may also induce time preference in consumption of the tempting good. To investigate

this aspect of behavior, we now permit the rate of willpower depletion to depend on the stock

of willpower remaining (fw < 0). Analysis of this case requires a more complete examination of

the consumer�s dynamic optimization problem (P1) since consumption need not be constant. As

noted above, problem (P1) is non-standard because of the discontinuity in the willpower depletion

process. To circumvent this di¢ culty, we examine a related �free endpoint�problem (P2), which

17Using similar arguments we can show that R and m are linked in the no smoothing regime as well.

11

is standard in its formulation and which has the same solution as problem (P1):

V (0) =

Z s

0e��tU [c (t)] dt+ e��sm (W (s))

subject to�R (t) = �c (t) (P2)�W (t) = �f (W (t) ; c (t))

R (s) � 0; W (s) � 0

R (0) = R � 0

W (0) =W � 0:

In problem (P2), the agent chooses not only an optimal consumption path but also a date s � Tbeyond which depletion of each stock ceases ( _W (t) = _R (t) = 0 for all t 2 (s; T ]). Problem (P2)

therefore allows all of the paths that are feasible in (P1) and evaluates each as (P1) does. However,

problem (P2) also allows additional feasible paths: those where willpower depletion ceases even

though cake remains (R(s) > 0). To establish that the solutions to the two problems are the

same, we need merely prove that these additional paths are never optimal in (P2). Intuitively,

this is obvious since one can always dominate such paths by consuming marginally more cake in

the neighborhood of s: This additional consumption is feasible since (by hypothesis) more cake is

available; moreover, consuming it would relax the willpower constraint.18

Working with problem (P2), then, the Hamiltonian is given by

H (c (t) ; R (t) ;W (t) ; � (t) ; � (t) ; t) = e��tU (c (t))� � (t) c (t)� � (t) f (W (t) ; c (t)) ;

we will refer to it as H (t) when no confusion arises. The �rst-order conditions include:

c (t) � 0, e��tU 0 (c (t))� � (t)� � (t) fc � 0 and c.s. (3)

_W (t) = �f (4)

_R (t) = �c (5)

_� (t) = 0 (6)

_� (t) = � (t) fW (7)

T � s � 0; H (s)� �e��sm (W (s)) � 0 and c.s. (8)

R (s) � 0; � (s) � 0 and c.s. (9)

W (s) � 0; � (s)�m0 (W (s)) � 0 and c.s. (10)

18For a formal proof of the equivalence of P1 and P2, see our Technical Appendix available online at

www.umich.edu/~dansilv/research.

12

Using this dynamic system, we can study the impact of willpower concerns on the agent�s time

preference of consumption. To isolate the e¤ects of the willpower constraint, we continue to assume

that there is no discounting (� = 0).

2.2.1 The Time Path of Consumption

To begin, consider the intuitive tradeo¤s revealed by the �rst-order conditions characterizing

optimal consumption at t: When consumption is strictly positive, we can rewrite condition (3) as:

U 0 (c (t))| {z }direct marginal bene�t

+ �(t)(�fc)| {z }indirect marginal bene�t

= �(t)|{z}marginal cost

: (11)

Grouped in this way, condition (3) shows that consuming at a slightly faster rate at time t generates

two marginal bene�ts and one marginal cost. The direct marginal bene�t (U 0 (c (t))) is the usual

one: an increase in utility at time t (expressed in utils at t = 0) that results from consuming

more at time t. Similarly, the marginal cost of consuming at a faster rate at t (�(t)) is also the

standard one; it re�ects the utility lost because the additional cake consumed at t can no longer be

consumed some other time. What is distinctive is that increasing consumption also has an indirect

marginal bene�t (�fc�(t)): This indirect bene�t derives from the fact that, by increasing the rate

of consumption, the agent depletes willpower at a slower rate (�fc) and each unit of willpowersaved at time t is worth �(t) utils. At an interior optimum, the sum of the two marginal bene�ts

must equal the marginal cost.

The optimality conditions also reveal an important di¤erence between the two depletable re-

sources, cake and willpower, when the rate of depletion depends on the stock of willpower remaining.

Additional cake is equally valuable whenever it arrives since its mere availability, before it is con-

sumed, provides no services and its future arrival can be anticipated by consuming more in advance.

That explains why the shadow value of additional cake is constant over time ( _�(t) = 0). In contrast,

when fW < 0 and the willpower constraint is binding, additional willpower is more valuable the

earlier it arrives because its mere presence provides a service: the more willpower one has available

at t, the less must be depleted over a short interval to restrain consumption by a given amount.

This is not a service upon which one can draw simply by recognizing that more willpower is avail-

able in the future. For this reason, the shadow value of willpower declines with time ( _�(t) < 0 from

(7) whenever fW < 0). This distinction between the two depletable resources disappears when

fW = 0. In that case, additional willpower� like additional cake� is equally valuable whenever it

arrives and, as we have seen, it is optimal to consume at a constant rate until the cake is exhausted

at s � T:19

19To see that c(t) must be constant for t � s, note that (4) implies that �(t) is constant. If it is assumed that

13

The Possibility of Negative Time Preference As we showed in subsection 2.1, when willpower

depletion depends only on the consumption level (fW = 0), it is optimal to consume at a constant

rate as long as cake remains. More generally, however, limited willpower may induce time-varying

paths of consumption. Indeed, willpower concerns may result in increasing paths of consumption

(negative time preference).

To see the sources of time preference in the willpower model, note that the sum of the direct

and indirect marginal bene�ts of increased consumption de�ned in (11) cannot vary over time,

since (6) insures that �(t) is constant. Hence, whenever the path of the indirect marginal bene�t

of increased consumption is increasing the direct marginal bene�t path must be decreasing. Given

the concavity of the utility function, this means that, in these circumstances, consumption must be

increasing. By a similar argument, a decreasing indirect marginal bene�t path requires a decreasing

consumption path.

The following proposition characterizes the (local) rate of change in consumption and thus gives

a condition for consumption to be locally increasing.

Proposition 1 Suppose c (t) > 0 for t � s: Then _c T 0 as � ddW

ddc ln f(W; c)) T 0:

Proof. Since � (t) is constant over time, di¤erentiating equation (11) with respect to t implies

[U 00 (c)� �fcc] _c+ f _�(�fc) + �fcW (� _W )g = 0: Because [U 00 (c)� �fcc] < 0; _c T 0 if and only if�fc�ffcW�fc (� _W )+

_��g T 0: Then, using (7) and (4) to replace

_�� and� _W , we obtain _c T 0 if and only if

�fffcW�fcfW g T 0: The result then follows because �fffcW�fcfW g T 0 as � ddW

ddc ln f(W; c)) T 0:

This proposition implies that, when strictly positive, consumption is constant in only three

circumstances: (1) when willpower is not scarce (� = 0), (2) when the rate of willpower depletion

is independent of the stock of willpower remaining (fW = 0), or (3) when, despite neither of the

preceding circumstances, the indirect marginal bene�t of a marginal increase in consumption is

constant. Thus, if willpower is scarce (�(t) > 0) and the rate of willpower depletion depends on

the stock (fW < 0), the prediction of a constant consumption path is a knife-edge case.

Proposition 1 can also be seen to say that when willpower is scarce, consumption strictly

increases whenever the logarithm of the depletion function has a strictly positive cross partial

derivative.20 Knowing this, it is straightforward to construct examples where the consumption

path will rise or fall monotonically before jumping to zero. These examples provide useful intuition

for optimal choice under a willpower constraint.

fW � 0 then fc(W (t); c(t)) does not vary with W (t) and, given (6), �(t) is constant. Since we are assuming that

� = 0; the solution to (3) will be the same for all t � s.20The reader may recognize this condition as logsupermodularity of the depletion function.

14

Consider a willpower depletion function f that is the product two functions, K (W ) and g (c).

The cross partial of the logarithm of such a function is zero, so the consumption path must be �at

prior to the last moment of consumption: If fcW > 0; however, simply adding a strictly positive

(respectively, strictly negative) constant A to any function of this form generates a strictly increasing

(respectively, strictly decreasing) consumption path. To see this formally let

f (W; c) = A+K (W ) g (c) with K 0 < 0 and g0 < 0:

Here, a given act of will is more depleting when the agent is already depleted. In addition, willpower

is either being drained (A > 0) or replenished (A < 0) at a constant rate.21 Note that,

d

dW

d

dcln f(W; c) =

K 0 (W ) g0 (c)A

[A+K (W ) g (c)]2

and thus, by Proposition 1

sign ( _c) = sign�K 0g0A

�:

Since g is strictly decreasing, if K 0 < 0 then _c T 0 as A T 0: So whenever the willpower is scarce,K 0 < 0, and A > 0, the path of optimal consumption must increase over time. The increasing

path emerges in this example from the interaction of two features. First, because A is positive,

willpower will disappear as time elapses; this is a �use it or lose it�situation. However, this feature

alone would not generate an increasing path of consumption; if K 0 were zero, the consumption

path would be �at. The consumer would anticipate the shrinking willpower stock and behave, in

that circumstance, exactly as he would with a smaller but unshrinking initial willpower stock. An

increasing path thus requires the second feature of this example: that willpower depletion depend

on the remaining stock of willpower. Because K 0 is negative, a given act of will is more depleting

when the willpower stock is low. So the agent optimally takes the opportunity to exert acts of will

when they have a relatively low opportunity cost (when W is large) and before time erodes the

willpower stock. This simple example thus shows how limited willpower may induce even negative

time preference in the absence of time discounting.

A preference for increasing paths of consumption has previously been attributed to utility from

beliefs (anticipation). Here we see how problems of self-control may also generate incentives to

�save the best for last�or �get the hard part out of the way �rst.�

21This exogenous draining or replenishing of willpower is importantly di¤erent from the endogenous depletion we

have been modeling all along and di¤erent from the endogenous replenishment modeled in section 4 where we allow

willpower to behave like a muscle �drained by exertion over the short term, but built by exertion over the longer

term.

15

While our model is �exible enough to explain both positive and negative time preference revealed

in consumption choices, it can be rejected. The problem of predicting the e¤ects of willpower on

consumption pro�les is analogous to predicting the e¤ects of price on demand or labor supply; in the

absence of additional information, income and substitution e¤ects make the net e¤ect ambiguous;

but with su¢ cient information about preferences, unambiguous predictions can be derived. Here, as

Proposition 1 shows, with su¢ cient information about the willpower depletion technology the model

makes strong predictions about the qualitative features of the consumption path. Importantly,

choice experiments can provide the requisite information. It is relatively straightforward to design,

for example, choice experiments that would identify whether the rate of willpower depletion depends

on the stock of remaining willpower.

2.2.2 Almost Never Smooth Even When Feasible

In section 2.1, we showed that perfect smoothing, the signature prediction of the standard model,

may not occur when fW = 0 and the marginal value of willpower allocated to alternative activities

is constant and large enough. In this subsection we derive necessary and su¢ cient conditions

for perfect smoothing to occur under more general assumptions (fW � 0 and m(W ) either zero

or strictly increasing and weakly concave). We �nd that if an agent lacks an alternative use of

willpower, a necessary and su¢ cient condition for him to smooth perfectly is simply that he have

enough willpower initially for such smoothing to be feasible. If, instead, willpower has alternative

uses, in almost all circumstances he will refrain from perfect smoothing no matter how large his

initial stock of willpower.

No Alternative Uses of Willpower We consider �rst the case where willpower has no al-

ternative uses and thus willpower bequeathed to the future has no marginal value. Since our

model coincides with the canonical model when the willpower constraint is slack (�(t) = 0), perfect

smoothing is optimal whenever feasible.

Proposition 2 Let WH be the minimum level of initial willpower such that setting c (t) = RT for

t 2 [0; T ] is feasible. Denote the optimal consumption path as c� (t). Let � = m (W ) = 0: Then ifW � WH the cake is exhausted, and c� (t) = R

T for t 2 [0; T ] : If instead W < WH then both the

cake and willpower are exhausted (R (s) =W (s) = 0), and � (t) > 0 for all t 2 [0; s] :

Proof. See Appendix.

Corollary 1 If � = m (W ) = 0 and W < WH then when consumption is strictly positive it is

strictly increasing (resp. constant, strictly decreasing) if and only if ddW

ddc ln f(W; c)) is strictly

positive (resp. zero, strictly negative).

16

Proof. This follows as a consequence of Propositions 2 and 1.

Remark 1 (Procrastination) Corollary 1 describes the path of consumption when strictly positive.

As we have seen, however, this description does not preclude intervals of zero consumption. Suppose

that the agent must spend a �xed amount of time to complete a project, and the cake represents the

remaining hours of leisure. Then he may work hard on the project at the beginning, slack o¤ as

time goes by, but then cram for some interval just before the deadline.

When Willpower Has Alternative Uses Proposition 2 shows that if there is no other use

of willpower, perfect smoothing is always optimal when feasible; and thus the standard model is

nested in ours. We now move on to investigate when perfect smoothing is optimal if willpower has

alternative uses. We assume throughout this subsection that devoting more willpower to controlling

alternative urges is always useful and �nd that perfect smoothing is almost never optimal even when

the cake is exhausted at T .

To see why the optimality of smoothing depends on the alternative uses of willpower, note that

if it is optimal to use some willpower to regulate consumption and the remainder to regulate other

urges, then it must be that willpower cannot be advantageously re-allocated between the two uses

(�(s) = m0(W (s)) by condition (10)). However, we have assumed that additional willpower devoted

to alternative activities is valuable (m0(�) > 0). Therefore �(s) > 0: Since this shadow value weaklydeclines over time (condition (7)) and is strictly positive when consumption ceases, it must also

be strictly positive previously. But then our previous result implies that perfect smoothing occurs

only if ddW

ddc ln f(W; c)) = 0:

We know from the analysis in subsection 2.1 that even if consumption is �at when positive,

perfect smoothing still may not occur. It may be optimal instead to consume at a constant but

rapid rate in order to exhaust the cake before T and save more willpower for the alternative

activities. Intuitively, this would occur if additional willpower devoted to regulating other urges

was su¢ ciently valuable.

How valuable must willpower be in alternative uses for perfect smoothing to fail even though

optimal consumption is �at whenever positive? For perfect smoothing to be optimal it must be

that, when c (t) = RT and s = T; the marginal return from allocating a bit more willpower to

alternative activities, m0(W (T )); is weakly less than the marginal value of additional willpower

allocated to consumption under perfect smoothing. We show below that this latter marginal value

can be bounded above by:

mH(W; c) =�U (c)� U 0 (c) c

�= (f (W; c)� fc (W; c) c) :22

22 If no smoothing occurs, then mH(W; c) = m0(W (s)). This generalizes equation (2).

17

Thus, if we denote by W the willpower available at T if perfect smoothing has been implemented,

and the rate of consumption under perfect smoothing by cH ;then if m0(W )) > mH(W ; cH); perfect

smoothing never occurs.23

The following proposition summarizes the necessary conditions derived above and, in addition,

provides su¢ cient conditions for perfect smoothing to be optimal.

Proposition 3 If additional willpower devoted to alternative activities is valuable (m0(�) > 0), andif the agent perfectly smooths consumption then d

dWddc ln f(W; c)) = 0, for c = cH andW 2

hW ; �W

i;

and m0(W ) � mH(W ; cH). Moreover, if ddW

ddc ln f(W; c)) = 0, for all c � cH > 0 and W 2

hW ; �W

iand m0(W ) � mH(W ; cH) then perfect smoothing must occur.24

Proof. We have already proved, in the text, that when m0(�) > 0 perfect smoothing requiresddW

ddc ln f(W; c)) = 0, for c = cH and W 2

hW ; �W

i: Thus, to prove the �rst statement, it remains

only to derive that m0(W ) � mH(W ; cH): If any willpower is used to restrain consumption, op-

timality of this decision requires that m0(W (s)) � � (s) : Next we �nd an upper bound on � (s)

using the �rst-order conditions. Condition (3) implies that U 0 (c (s)) � � (s) fc � �: Multiplying

both sides by c (s) we get, �U 0 (c (s))� � (s) fc

�c (s) � �c (s) :

Condition (8) requires

H (s) = U (c (s))� �c (s)� � (s) f (W (s) ; c (s)) � 0:

Combining these two inequalities we see that

� (s) ��U (c)� U 0 (c) c

�= (f (W; c)� fc (W; c) c) = mH(W; c):

To prove that the speci�ed conditions are su¢ cient for perfect smoothing to be optimal, assume

the contrary. Since the conditions imply that consumption must be constant when positive,

perfect smoothing could only fail because s < T . If consumption terminates at s < T; then

c =�Rs > cH : Since less restraint will be exercised over a shorter interval W (s) > W > 0: This in

turn has two implications. Since more willpower will be bequeathed to the alternative activities,

m0(W ) � m0(W (s)): In addition, since W (s) > 0; (10) implies that � (s) = m0 (W (s)) > 0: By

hypothesis, s < T: Then, as shown above, the �rst-order conditions imply �(s) = mH(W (s); c):

23Clearly W depends on the initial levels of willpower and cake but we suppress this dependence for simplicity.24Recalling the case studied in Section 2.1, where f (W; c) = g (c) and m0 (�) = m; this condition reduces to

m � (U (cH)� cHU 0 (cH)) = (g (cH)� cHg0 (ch)).

18

Since mH(�; �) is weakly increasing in the �rst argument and strictly increasing in the second ar-gument, mH(W (s); c) > mH(W ; cH): Given the hypothesis that s < T we therefore conclude that

m0(W ) � m0(W (s)) = �(s) = mH(W (s); c) > mH(W ; cH): But m0(W (s)) > mH(W ; cH) violating

m0(W ) � mH(W ; cH).

Thus Proposition 3 shows that when willpower has alternative uses, the necessary conditions

for perfect smoothing are restrictive, but not impossible to meet. Limited willpower largely undoes

the signature prediction of the standard theory of intertemporal choice. If willpower is depleted by

restricting consumption in the right way, and if the marginal value of willpower allocated to other

activities is not too high, then the consumer may still smooth perfectly.

3 Comparison with Alternative Models of Self-Control

The preceding sections were largely focused on how the predictions of a willpower model di¤er

from those of a standard model of intertemporal choice. In this section we compare our willpower

model with existing models of self-control.

We note �rst that limited willpower provides complementary explanations for the hallmark

predictions of self-control models: a preference for commitment, profound procrastination, and

apparent time-inconsistencies. Consider �rst a taste for commitment. As in each of the leading

models of self-control, an agent in our model would strictly prefer to have his �cake�or paycheck

doled out to him by a savings club.25 For if the entire amount were available, resisting spending

it would deplete his willpower and willpower has a scarcity value. Next consider profound pro-

crastination. Like other models of self-control, limited willpower also explains extreme forms of

procrastination. Suppose, for example, an agent must spend a �xed amount of time on an assign-

ment before a certain deadline, but can allocate the remaining leisure time optimally.26 Then if the

agent is willpower constrained, he may enjoy leisure early in the program, and later work non-stop

until the deadline.

Depletable willpower is also consistent with apparent intertemporal preference reversals. When

25Benabou and Tirole (2004) is an exception, as in that model commitments may prevent the agent from obtaining

valuable information about his willpower.26This is exactly the setup Fischer (2001) used to investigate procrastination but we have introduced a willpower

depletion constraint. Without it, Fischer �nds that the discount factor required to explain procrastination in a model

with standard time discounting and additively-separable utility is unrealistic. Moreover, with standard time discount-

ing, such procrastination is not something to be avoided; even if they could, agents would not seek commitments to

better smooth their work e¤ort. Our model can generate procrastination (zero consumption of leisure for the �nal

phase of the planning horizon) for any discount rate, and agents in our model would, if possible, seek commitments

to avoid it.

19

asked to choose between a smaller reward that will arrive immediately and a larger one that would

arrive after some delay, an agent with limited willpower may choose the former. The willpower

cost of resisting the immediate, smaller reward may outweigh the utility gain from receiving the

later, larger reward. However, if asked to choose now one of these same two options set o¤ in the

temporal distance, the consumer may choose the later, larger prize. The choice now of a prize to

be delivered in the future allows the consumer to commit irreversibly to an option. He then prefers

the later, larger prize since willpower ceases to be required to resist the earlier, smaller prize. In

this way a willpower-constrained agent may appear to behave in a time-inconsistent manner.

While it o¤ers complementary explanations for a taste for commitment, preference reversals

and profound procrastination, the willpower model also has implications that distinguish it from

existing models of self-control.

The distinguishing feature of the limited-willpower model is the linkages that it generates be-

tween otherwise unrelated choices. We saw the implications of these linkages in the �no smoothing�

results, domain-speci�c time preference, spillovers of self-control problems into demand for non-

tempting goods, and poverty a¤ecting patience. Perhaps most basically, we also saw an implication

of these links as consumption of even non-addictive goods in the past had in�uence on future choices

regarding seemingly unrelated goods. Thus, di¤erent from models of hyperbolic discounting (Laib-

son 1997, O�Donoghue and Rabin, 1999), and temptation costs (e.g., Gul and Pesendorfer 2001,

2004), keeping the current choice set constant, in our model the history of past choices matters for

behavior.

By linking otherwise unrelated, and non-addictive self-control activities, and by explicitly mod-

eling a willpower reserve, our model can be distinguished from models of cues like Laibson (2001)

and Bernheim and Rangel (2004). And even if attention is restricted to the e¤ects of past self-

restraint in one activity on future self-restraint in the same activity, there remains a di¤erence. In

the cue models, past e¤orts at self-restraint make it less likely that one will enter a hot state and

thereby lose all self-control. In our model, by contrast, the past exercise of self-restraint will, at

least over the short term, make future self-control less likely.

Benabou and Tirole (2004) o¤er an alternative model of willpower. In their model, agents

are quasi-hyperbolic discounters, but have limited knowledge of the degree of their present-bias

(willpower for Benabou and Tirole). In the moment of greatest temptation agents know their

willpower, but later can only infer their preferences from previous experience. To build self-

reputation and show their future selves that they are not impatient, agents apply self discipline.

The mechanism through which willpower works in their model is thus quite di¤erent from ours,

and would in general lead to di¤erent predictions. One important di¤erence is in how various

self-control activities would be linked. In their model, an agent who successfully uses self-restraint

20

in one domain would become increasingly con�dent in his willpower and, thus, more likely to use

self-restraint in another domain. In our model, the opposite is true, an agent who is depleted

from using willpower in one situation will be less likely to use self-restraint in another seemingly

unrelated situation.

Loewenstein and O�Donoghue (2005) and Fudenberg and Levine (2006) also have related models

of costly self-control. In Loewenstein and O�Donoghue�s framework the past exercise of self-restraint

a¤ects the direct cost of self-control in the future. Their model is quite general and they apply it

both to intertemporal choice and to static choice under uncertainty. We view our paper as comple-

menting theirs with a further examination of the implications of limited willpower for intertemporal

choice and with greater emphasis on the allocation of willpower across competing demands for self-

restraint. Fudenberg and Levine (2006) model a game between a long-run self and a sequence

of short-run (myopic) selves. The short-run selves are both the �doer� selves (they �physically�

make the choices) and the conduits of �ow utility to the long-run self. At a cost, the long-run

self may alter the preferences of the short-run self and thereby exercise self-control. By allowing

cognitive loads to a¤ect the cost of a given act of self-control, Fudenberg and Levine (2006) can

accommodate the in�uence of such loads on the willingness to exercise self-control within a period.

They also discuss how within period linkages between activities requiring self-control may emerge

if self-control costs depend on multiple activities in a non-linear way. Our model is di¤erent in that

it predicts longer-term e¤ects, through the willpower constraint, of both shocks to willpower stocks

and anticipated, alternative needs for self-restraint.27

In addition, the willpower model�s accommodation of increasing paths of consumption distin-

guishes it from the bulk of existing models of self-control in which agents are tempted to over-

consume.28 Negative time preference is usually viewed as unrelated to self-control and is attributed

to utility from beliefs (anticipation). Models of anticipatory utility can accommodate increasing

consumption paths, but cannot by themselves explain a taste for commitment, or profound procras-

tination, or links between seemingly unrelated acts of self-regulation. As we have shown, optimal

willpower management can explain all of these phenomenon while, at the same time, inducing a

preference for �saving the best for last.�

Note that our analysis shows how willpower concerns may induce a preference for increasing

consumption in the absence of commitment. Our willpower model predicts perfect smoothing of

27There is also a conceptional di¤erence between our approach and that taken by both Loewenstein and O�Donoghue

(2005) and Fudenberg and Levine (2006) in that we model self-control costs as opportunity, rather than direct costs.

See the end of this section for more on this issue.28The exception is Benabou and Tirole (2004). A complete discussion of this issue is left to the Technical Appendix

available online at www.umich.edu/~dansilv/research.

21

consumption if the consumer can commit to that path at any time prior to the moment consumption

begins. The importance of commitment thus permits a simple way to distinguish the willpower

motive from anticipatory utility in contributing to tastes for increasing consumption. To the extent

that such tastes are revealed under commitment, it points to anticipatory utility from beliefs. To

the extent that such tastes are enhanced by the absence of commitment, it points to willpower

concerns.

In this regard, there is suggestive �eld evidence of substantial increases in consumption ex-

penditure over short time intervals by unambiguously uncommitted consumers. Studies of high

frequency data on individual expenditures in Britain indicate that consumers sometimes choose in-

tervals of increasing consumption between the arrival of paychecks.29 Both Kelly and Lanot (2002)

and Hu¤man and Barenstein (2004) �nd that the non-durable expenditures of British consumers

exhibit a U-shaped pattern over the course of the four weeks between paychecks. (See, for example,

their Figures 1 and 4, respectively). Kelly and Lanot (2002) attribute the U-shaped pattern of

consumption expenditure to uncertainty and precautionary saving within the pay period. Hu¤-

man and Barenstein (2004) attribute the interval of apparently increasing average consumption to

measurement error.30

Finally, we conclude this section by distinguishing our treatment of self-control as an endogenous

shadow cost from the treatment in the previous literature as an exogenous direct cost.31 Presumably

29There is also some, less statistically certain evidence of intervals of increasing consumption in high frequency

data on daily expendiure by US consumers. See Stephens (2003), Tables 2 and 3 for total expenditure and Figures

4c-4g for non-durable expenditures.30The Family Expenditure Survey used in both studies asks respondents to record their expenditures in a diary for

two weeks. The day upon which the diary begins is randomly assigned. Public use data aggregate each household�s

expenditures by week. Researchers can identify whether a respondent is paid monthly and the date his last paycheck

arrived but not, precisely, the date of his next check. Therefore, diary weeks that begin 25 days or more after the last

check arrived will typically include expenditures from at least one day after the next check has arrived. If consumption

is higher on days just after the check arrives, this measurement error may explain the higher levels of consumption

in diary weeks that began 25-30 days after the last check arrived (see Hu¤man and Barenstein, 2004, Figure 4). This

error would not seem to explain, however, the increase in average expenditure between weeks starting 21 (perhaps

17) and 24 days after the last check. Neither can this measurement error explain why average expenditure is often

higher in the third week after the arrival of the last check than it is in the second (Kelly and Lanot, 2002, Figure 1).31 In discussing an appropriate formulation to capture Baumeister�s experiments, Loewenstein and O�Donoghue

(2005) regard the exertion of willpower as generating disutility although Loewenstein (2000) regards the matter as

an open question. Whether or not willpower depletion has a utility cost seems di¢ cult to resolve empirically. What

we have resolved analytically, however, is that including the stock of willpower (or its rate of change) in the utility

function is not necessary to capture either the behaviors psychologists have documented in their laboratories or to

explain a number of prominent �anomalies�behavioral economists have reported in the �eld. For these purposes, it

is su¢ cient to append to the conventional formulation of intertemporal utility maximization the additional constraint

22

there are circumstances in which the classical model predicts well. It is, therefore, an attractive

feature of any new model supplanting a classical one that it identi�es the circumstances in which the

classical results arise endogenously. If willpower depletion is appended as a constraint, there will be

circumstances in which perfect smoothing will be optimal and consumption behavior predicted by

the willpower model will be observationally equivalent to the one predicted by the classical model.

4 Extensions

Our approach in the previous sections has been to examine the implications of limited willpower

in the simplest intertemporal problem. For the sake of simplicity we have also assumed that the

agent always knows his stock of willpower with certainty and that willpower, if it is renewable,

recovers at a rate unrelated to how it is exercised. In this section, we discuss the consequences of

relaxing each assumption.

4.1 Depleting a Stock of Willpower of Unknown Size

Loewenstein (2000) points out that an agent may not be a good judge of the willpower he has

in reserve. A similar issue has been analyzed by economists studying optimal consumption of a

cake of unknown size.32 That analysis can easily be adapted to the case where the agent knows his

cake size but not his initial reserves of willpower. As we show, along the optimal path the agent

may discover either that he overestimated his self-control resources or that he has more willpower

than he anticipated and has �a second wind.�

A two-state example will clarify the basic ideas. Suppose an agent has one of two initial stocks of

willpower and assigns positive probability to each state. Assume that the agent has no alternative

use of willpower and that even the larger stock of willpower is insu¢ cient to implement perfect

smoothing. As in subsection 2.1, assume that fW = 0 so that consumption is constant when strictly

positive.

If the agent knew that his willpower stock was low, he would conserve it by consuming at a

rapid, constant rate and would deplete his cake quickly, after which he would have nothing left to

consume until the end of the horizon. Denote as sl the date when he would exhaust his cake if he

was sure his initial willpower stock was low. Similarly, denote as sh > sl the date when he would

exhaust his cake if he was sure his initial willpower stock was high. Suppose, however, that the

consumer could not observe his willpower reserves directly. Then, whatever consumption path he

that the consumption path chosen must not overexhaust the agent�s willpower.32See Gilbert (1979), among others.

23

chooses, he would obtain no information about his initial willpower stock until the date when his

willpower would have run out if his initial stock had been small. At that point, he could infer the

size of his initial reserves: if he loses control because he has no more willpower, then his initial

reserves were small; if he retains control because he has more willpower, then his initial reserves

were large.

Under uncertainty of this kind, it is optimal to consume at a constant rate intermediate be-

tween the alternative consumption rates he would choose under certainty. Since this would involve

restraining his consumption more than is optimal if he were certain his willpower stock was low,

he would run out of willpower sooner in the event that his is initial stock was in fact low. If we

denote as sul the date when he would run out under uncertainty if his initial stock of willpower was

in fact low, then sul < sl. In that event, lacking willpower reserves to control his consumption, he

would exhaust the remaining cake at the rate �c and then would have nothing left to consume until

the end of the horizon.

On the other hand, if at sul the agent discovered that he still had the ability to restrain

consumption, then he would rationally conclude that his initial willpower stock had been high.

Since he had been consuming at a faster rate than if he had known this state from the outset, he

would �nd himself at s+ul with more willpower and less cake. In this circumstance, the consumer

would take advantage of this �second wind�by restricting his consumption below the constant rate

he would have chosen if he had known from the outset of his high reserves of willpower.

4.2 Self-Control Builds Willpower Like a Muscle

Common intuition and some psychology experiments indicate that willpower may be like a

muscle: controlling visceral urges depletes willpower over the short term, but the regular exercise

of self-restraint may eventually build willpower. Most important for our purposes here, there is

some evidence that willpower can be built up in one domain and then used to advantage in other

arenas (Muraven et al., 1999 and Muraven and Baumeister, 2000).33

To incorporate the e¤ects of buildable willpower, we introduce a third state variable, muscle,

the level of which is denoted by M (t) : We augment our earlier model in two ways. First, we

alter the transition equation for willpower (equation 4) to: _W (t) = M (t) � f (W (t) ; c (t)) : As

before, willpower is depleted by restraining consumption, f (W (t) ; c (t)) ; but this depletion is

moderated by the service �ow from the stock of muscle, M (t) : Since the stocks of willpower

33 In one experiment (Muraven et al., 1999), subjects who participated in two-week self-control drills (regulating

moods, improving posture, etc.) later showed signi�cant increases in the length of time the would squeeze a handgrip

relative to those who did not participate in the drills.

24

and muscle cannot jump, the only way to alter the rate of willpower depletion immediately is by

altering contemporaneous consumption. However, current exercise helps develop future muscle and

this muscle provides additional willpower at rate : If = 0; we recover our original transition

equation. Second, we assume that the rate at which muscle develops (or deteriorates) is given by_M = f (W (t) ; c (t)) � �M (t) : The idea is that exercising willpower today contributes to one�s

future muscle but the contributions decay.

The optimal consumption path in this muscle model shares some qualitative features with that

in the model without muscle. First, the cake is entirely consumed. Second, if willpower has no

alternative uses, perfect smoothing is optimal whenever it is feasible. In particular, for every initial

level of muscle there is a willpower levelW ~H above which the optimal path entails perfect smoothing.

This is true because, for every initial stock of muscle there is an initial stock of willpower su¢ ciently

large such that perfect smoothing is feasible and therefore optimal.34 If we start with an initial

level of willpower su¢ cient for perfect smoothing, decreases in that stock will eventually lead to a

willpower level�W ~H

�where any further reduction in the initial stock of willpower will make the

perfectly smooth path infeasible.

As in the model without muscle, on the optimal path the sum of the direct and indirect marginal

bene�ts of increased consumption must remain equal to the marginal cost of that consumption

(having a bit less cake for the future). But now, expanding consumption has a second marginal

cost: besides depleting cake it reduces future muscle mass.

The opportunity to build muscle a¤ects the optimal consumption path. For example, in the

case where fW = 0, it was optimal to consume at a constant rate until the cake was exhausted in

the absence of muscle-building. In the case where muscle decays at a faster rate than it contributes

to willpower (� � ); consumption is always increasing. Relative to the optimal path in the absenceof muscle, the ability to build willpower through its exercise leads the agent to bear down at the

beginning of the program in order to enjoy a greater willpower later. A more detailed analysis of

this case is available in an online appendix.35

5 Conclusion

This paper has explored the consequences of including in a conventional model of intertemporal

choice a cognitive constraint consistent with a large body of experimental evidence. Speci�cally,

we assumed that if an agent restricts his consumption of a tempting good, then exercising self-

34 Indeed, if the initial muscle level is large enough, the agent will be able to achieve perfect smoothing without any

initial willpower.35The online appendix may be downloaded from www.umich.edu/~dansilv .

25

restraint depletes his �nite stock of willpower �a resource useful for regulating urges of all kinds.

This willpower constraint captures the common notion, consistent with laboratory experiments,

that an individual has a limited, though positive, capacity to control his own impulsive behaviors.

To study how a willpower constraint a¤ects choices over time, we introduced it in the simplest

intertemporal model, where an agent decides how to consume a cake (or paycheck or stock of leisure

time) over a �xed time period. Any willpower left over after regulating intertemporal consumption

is used to control urges in other activities.

The model is simple but generates a rich set of implications. Most important, a willpower

constrained consumer regards seemingly unrelated activities as linked because he uses the same

cognitive resource to exercise self control in di¤erent activities. As a consequence, prior acts of

self-restraint may a¤ect an agent�s subsequent conduct because they reduce the stock of willpower

he can use to regulate his behavior. The link between seemingly unrelated behaviors also implies

that agents with limited willpower may appear to have domain-speci�c time preference �behaving

as if completely myopic with regard to some choices and extremely forward-looking with regard to

others. As important, time preference in one domain will change with the returns to self-control

in another. Limited willpower also implies that self-control problems will a¤ect demand for even

nontempting goods, and that a lack of resources may cause impatience.

Limited willpower also has implications for the time-path of consumption. We saw that, because

he draws on a single cognitive resource to regulate various urges, an agent may never smooth

consumption, even if his initial willpower makes it feasible. In addition, an agent in our model may

increase his consumption over time because exercising self control later, when his stock of willpower

is reduced, requires more willpower than exercising the same self control earlier.

We considered two extensions of the intertemporal model. First, we considered the possibility

that the decision maker is uncertain about his stock of willpower. We found that the agent would

optimally consume at a higher rate as a hedge against the risk of running out of willpower and, if

he learns that he has larger willpower reserves, experiences a �second wind�that permits further

self-discipline later on the optimal path. Second, we investigated what would happen if current

exercise of self control, while immediately depleting willpower, also builds willpower reserves in

the future. We found that renewable willpower induces a time preference even when consumption

would have been constant if willpower had been nonrenewable.

That willpower is scarce and depletable accords with both introspection and experiment. To

explore how this constraint a¤ects intertemporal behavior, we appended it to the canonical model

of intertemporal choice. To our surprise, we found that the augmented model accounts both for

prominent anomalies of intertemporal choice that have been the focus of the self-control literature

and for other anomalies that are often treated as altogether separate phenomena requiring separate

26

models. Future research should clarify experimentally the form of the willpower depletion function.

Given that taking account of a willpower constraint is so tractable, it should then be embedded in

other economic models where self-control issues arise.

6 Appendix

Proof of Proposition 2: From (6) � (t) is a constant function, and with a slight abuse of

notation we denote this constant as � � 0: First assumeW �WH : Consider the case where � (t) = 0

for all t 2 [0; s] : Since U 0 (�) > 0; (3) requires � > 0 and c (t) constant. With a slight abuse of

notation, we denote this constant as c � 0: Since � > 0; (9) requires R (s) = 0: Since R (0) = R > 0;then c > 0: From (8), T � s � 0. If T � s > 0; (8) would require U (c) � U 0 (c) c = 0: But since

U 0 (�) > 0; U 00 (�) < 0; and c > 0; U(c)� U 0(c)c > 0; hence, T � s > 0 must be ruled out. It followsthat when � (t) = 0 for all t 2 [0; s] ; then s = T and c = R

T : By de�nition of WH ; W (T ) � 0: Thus(10) is satis�ed. This proves the �rst statement in the proposition. Now assume W < WH : Then

� (t) > 0 for all t 2 [0; s] ; for suppose to the contrary that � (t) = 0 for some t 2 [0; s] : Equation(7) implies that � (t) is weakly decreasing and can be written as � (t) = � (0) e

R tn=0 fW (W (n);c(n))dn:

Since eR tn=0 fW (W (n);c(n))dn > 0 for all t; � (t) = 0 for some t 2 [0; s] ; implies that � (t) = 0 for all

t 2 [0; s] : But as seen above the conditions above then imply that c = RT which is infeasible when

W < WH : So if W < WH then � (t) > 0 for all t 2 [0; s] ; and by (10) W (s) = 0: To satisfy (3)

with U 0 (�) > 0 and ��fc > 0 requires � > 0; and, again, since � > 0; (9) requires that the cake isentirely consumed (R (s) = 0) :

27

Figures

m

( )sWs,

( )sW

s

Perfect smoothing region No smoothing region

Figure 1: Optimal time (s) to exhaust the cake and optimal willpower bequest (W (s)) as a function

of the return to willpower applied to alternative activities (m) :

28

s,W s( )

( )sW

s

R

Perfect smoothing regionNo smoothing region

Poor Rich

Figure 2: Optimal time time (s) to exhaust the cake and optimal willpower bequest (W (s)) as a

function of the initial cake size (R):

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31